# Families of complex spaces and the foundations of analytic geometry

*November 1960*

#### Translator’s note

*This page is a translation into English of the following:*

Douady, A. *Séminaire Henri Cartan* **13 (1)** (1960–61), Talks no. 2 (“Variétés et espaces mixtes”), 3 (“Déformations régulières”), and 4 (“Obstruction primaire à la déformation”). `http://www.numdam.org/item/SHC_1960-1961__13_1`

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `367a5c6`

*[Translator] According to the complete list of talks, the notes from the first talk of the 1960/61 Séminaire Henri Cartan — “Fibrés en tores complexes” (also given by Adrien Douady) — were not copied, and thus seem to be lost to the past. What follows is a translation of the next three talks in this seminar series.*

# I. Category of models

Let

- the diagram
\begin{CD} U @>f>> U' \\@V{\pi_1}VV @VV{\pi_1}V \\B @= B \end{CD} commutes, where\pi_1 denotes the projection ofB\times\mathbb{C}^n toB ; and - for all
x\in B , the mapf_x\colon U_x\to U'_x is holomorphic, whereU_x = \{z\in\mathbb{C}^n \mid (x,z)\in U\} (and similarly forU' ).

If

More generally, if *morphism of \mathscr{S}_{f_1}* is a continuous map

- the diagram
\begin{CD} U @>f>> U' \\@V{\pi_1}VV @VV{\pi_1}V \\B @>>{f_1}> B' \end{CD} commutes; and f_x\colon U_x\to U'_{f_1(x)} is holomorphic for allx\in B .

If

# II. The definition of mixed spaces and mixed varieties

## 1. First definition

Let *mixed space* over

The structure thus defined is that of a * (\mathscr{C}^0,\mathbb{C})-mixed space*.
If

*. In this case,*\mathbb{C} -analytic mixed space

If * (\mathscr{C}^\infty,\mathbb{C})-mixed manifold* (resp.

^{1}

Let *morphism from V to V' over f_1* is a continuous map

## 2. An equivalent definition

We now give another way of defining mixed spaces, equivalent to the above.

Given separated spaces *pre-mixed space* consists of the structure of a *morphism of pre-mixed spaces over f_1* is a continuous map

A *mixed space* is a pre-mixed space *full subcategory*.

## 3. Deformations

A mixed space *proper* if

Let * \mathbb{C}-analytic deformation of V_0 over (B,b_0)* consists of a proper

The goal of this seminar is the study, at least local, and an attempt at a classification of,

Let * \mathbb{C}-analytic deformation (\pi\colon V\to B,i\colon V_0\to V) of V_0* is said to be

*locally complete*if, for any other deformation

*locally universal*is furthermore the germ of

It seems that every compact

# III. Vector fields

## 1. Study on models

Let

A holomorphic field of tangent vectors on *vertical holomorphic field* on *vertical holomorphic field on U* is a continuous (resp. …) map

*transport*f_*\theta of \theta by f is defined by

Now suppose that *projectable holomorphic field* if *projection* of the field *Zariski* tangent space to

If

A *projectable holomorphic field on U* is a

## 2. Vector fields on a mixed manifold

Let

- vertical holomorphic fields on an open subset of a fibre;
- vertical holomorphic fields on a open subset of
V ; - projectable holomorphic fields on an open subset of a fibre; and
- projectable holomorphic fields on an open subset of
V .

Let

e^\xi(t_1+t_2,y) = e^\xi(t_1,e^\xi(t_2,y)) , with the left-hand side being defined whenever the right-hand side is; and\frac{\partial}{\partial t}e^\xi(t,y)|_{0,y} = \xi(y) .

Note that

For

The proof is left to the reader.

# IV. The Spencer–Kodaira map

Let

\Theta_0 : the sheaf of germs of vertical holomorphic fields onV_0 ;\Pi_0 : the sheaf of germs of locally projectable holomorphic fields onV_0 ; and\Lambda_0 : the sheaf\pi^*T_0 , i.e. the sheaf of germs of locally constant maps fromV_0 toT_0 .

We have an exact sequence of sheaves on

The *Spencer–Kodaira map* is the composition

This map is an essential tool in the local study of deformations of

It is clear that, if the given mixed manifold is trivial (i.e. if

# I. The map \widetilde{\rho}

All throughout this talk,

Let

For every open subset

For the proper mixed manifold

*Proof*. —

*(Necessity).*If\pi\colon V\to B is locally trivial atb_0 , then, for every open subsetU ofB over whichV is trivial, we have\widetilde{\Pi}=\widetilde{\Lambda}\oplus\widetilde{\Theta} onV_U , and so\delta\colon\mathrm{H}^0(V_U;\widetilde{\Lambda})\to\mathrm{H}^0(V_U;\widetilde{\Theta}) is zero.*(Sufficiency).*Let(\eta_1,\ldots,\eta_p) be\mathscr{C}^\infty vector fields (resp. …) on a neighbourhood ofb_0 inB , such that(\eta_1(b_0),\ldots,\eta_p(b_0)) forms a basis of the tangent spaceT_0 toB atb_0 . It then follows from the hypothesis that the map\mathrm{H}^0(V_0;\widetilde{\Pi}) \to \mathrm{H}^0(V_0;\widetilde{\Lambda}) is surjective. So let(\xi_1,\ldots,\xi_p) be projectable holomorphic vector fields on a neighbourhood ofV_0 inV , that project to(\eta_1,\ldots,\eta_p) . Letf be the map defined on a neighbourhood of\{0\}\times V_0 in\mathbb{R}^p\times V_0 (resp.\mathbb{C}^p\times V_0 ) byf(t_1,\ldots,t_p,y) = e^{\xi_1}(t_1,e^{\xi_2}(\ldots,e^{\xi_p}(t_p,y)\ldots)). It follows from the proposition stated in [2, Section III.2] thatf induces an isomorphism of mixed manifolds fromU\times V_0 to\pi^{-1}(f_1(U)) overf_1 , whereU is a sufficiently small cubical neighbourhood of0 in\mathbb{R}^p , andf_1 is the map fromU toB defined byf_1(t_1,\ldots,t_p) = e^{\eta_1}(t_1,e^{\eta_2}(\ldots,e^{\eta_p}(t_p,b_0)\ldots)), which proves the theorem.

# II. The regular case

For all

For every open subset

We say that the proper mixed manifold *regular* if

- the dimension of
\mathrm{H}^1(V_b;\Theta_b) does not depend on the pointb\in B ; and - we can endow
E=\bigcup_{b\in B}\mathrm{H}^1(V_b;\Theta_b) with the structure of a\mathscr{C}^\infty vector bundle (resp. …) such that\widetilde{\varepsilon} is an isomorphism from the sheaf\mathrm{R}^1\pi_*\widetilde{\Theta} to the sheaf of germs of\mathscr{C}^\infty sections (resp. …) of the bundleE .

In fact, Kodaira and Spencer have shown [7] that, by identifying the

Then Theorem 1 has the following corollary:

For the proper mixed manifold

Indeed, since

At the end of this talk, we will construct a counter-example which shows that it is necessary to assume that the mixed manifold is regular.

# III. An example of non-regular deformation: Hopf manifolds

## 1. Hopf manifolds

Let *Hopf manifold defined by b*, is a compact

Note that

Let

We can identify

*Proof*. If

We have the exact sequence

*Proof*. The only thing that we need to verify is that the map

Now, to finish the proof of Proposition 2.
From Lemma 1, we have the following exact sequence:

Now

## 2. Mixed manifolds whose fibres are Hopf manifolds

Let

Note that the dimension of

## 3. Calculation of \rho

We have

The Spencer–Kodaira map

*Proof*. Let

Let

Set

## 4. A counter-example

Take

We can also see that

Let

However, the mixed manifold

## 5. Question (K. Srinivasacharyulu)

We know that the Hopf manifolds are non-K"{a}hler, and thus non-algebraic.
For

# Appendix

With the notation of §III.1, let

# Introduction

Let *deformation vector*.
We will give a necessary condition for

# I. Exact sequences of sheaves of algebras

Let *cup product*

A *sheaf of algebras* on

If

Let

*Proof*. Let

# II. The primary obstruction

Let

Let

Let

*Proof*. *(Proof of the Corollary).*
This is simply a particular case of Theorem 1;
note that

*Proof*. *(Proof of Theorem 1).*
Consider the following sheaves on

\Theta_0 : the sheaf of germs of vertical holomorphic fields onV_0 ;\widetilde{\Theta}_0 : the sheaf of germs of vertical holomorphic fields onV ;\Pi_0 : the sheaf of germs of locally projectable holomorphic fields onV_0 ;\widetilde{\Pi}_0 : the sheaf of germs of locally projectable holomorphic fields onV ;\Lambda_0 : the sheaf\pi^*T_0 , whereT_0 is the tangent space ofB atb_0 ; and\widetilde{\Lambda}_0 : the sheaf\pi^*\widetilde{T}_0 , where\widetilde{T}_0 is the space of germs atb_0 of fields onB of tangent vectors ofB .

We have the following diagram:

Let

—

- We make essential use of the fact that
\epsilon\colon\widetilde{T}_0\to T_0 is surjective, and thus of the fact thatB has no singularities. - We actually have
[\rho(u)\smile b]=0 for allu\in T_0 , for any classb\in\mathrm{H}^1(V_0,\Theta_0) that is in the image of\mathrm{H}^1(V_0,\widetilde{\Theta}_0) under\epsilon . In particular, for an elementa\in\mathrm{H}^1(V_0,\Theta_0) to be a regular deformation vector (in the sense of [Talk no. 3]), it is necessary and sufficient for[a\smile b]=0 for allb\in\mathrm{H}^1(V_0,\Theta_0) .

If *primary obstruction* to the deformation of *obstructions*, with ^{2} of *this* condition is sufficient.
Kodaira, Spencer, and Nijenhuis [5] have shown that, if

# III. An example of obstruction

## 1. The manifold V_0

Let

## 2. The mixed space V

In this example, every element of

There exists a mixed space

\pi^{-1}(b_0)=V_0 (the manifold defined in §III.1);- there exists an isomorphism
\sigma from a\mathbb{C} -analytic spaceB to the cone of elements\varphi\in\mathrm{H}^1(V_0,\Theta_2) such that[\varphi\smile\varphi]=0 ; and - for every subspace
B' ofB that has no singularities atb_0 , the Spencer–Kodaira map\rho from the tangent space ofB' atb_0 to\mathrm{H}^1(V_0,\Theta) agrees with\sigma\colon B'\to\mathrm{H}^1(V_0,\Theta_2) .

Let

Let

## 3. Calculating \rho_0

Let

Let

Let

# I. Definition of obstructions

## 1. The sheaf of germs of vertical automorphisms

Let

\pi_1\gamma=\pi_1 is the projectionB\times V_0 toB ;\gamma is the identity on\{b_0\}\times U .

Then

It is clear that

We can identify

Recall that a deformation germ of

*Proof*. *(Proof of Proposition 1).*
Let

Set *associated to the deformation*.
It will still be associated to the deformation if pass to a finer cover.
Let

Conversely, suppose we have a locally finite cover

Finally, we can show that all the above does indeed define a bijection between the set of local classes of deformations of

## 2. Higher obstructions

For every open subset *the deformation in one parameter*), for all

Now, if

The exact sequence of non-abelian groups
*necessary* condition for

Let *obstruction of order k of the element q* to be the direct image in

*trivial*if the identity element belongs to this subset. Being trivial is a necessary and sufficient condition for

If

This definition is used most of all in the case of deformations in one parameter (

# II. Calculation of obstructions

## 1. Relation to the sheaf \Omega

From now on, we work in the case of deformations in one parameter, i.e.

Let

Then

*(Campbell–Hausdorff).*
We can identify

The proof of this proposition will not be given here.
We denote by

## 2. Calculation of the primary obstruction

Now let

Note that, if we denote by

Consequently,

## 3. Calculation of the secondary obstruction

Now suppose that

This cocycle can be lifted to *Massey triple product*

We can try to calculate this secondary obstruction without leaving the sheaf

## 4. Using spectral sequences

Let

However, if

*Proof*. Let

This proposition allows us to construct non-trivial examples of secondary obstructions.
Consider the group

I do not know of any examples of non-trivial secondary obstructions on varieties

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